Vector Differentiation

     As with functions of one variable we define the derivative as the limit of a difference quotient.

Definition (Difference Quotient) The difference quotient of a vector function is the vector function:



where is a change of the variable

Definition (Derivative of a Vector Function) The derivative of a vector function is the vector function defined as the limit



provided this limit exists.

Proposition (Derivative of a Vector Function) The vector function



is differentiable whenever the component functions , , and are each differentiable and in this case

    Proof. By the definition of the derivative of a vector function,





Using properties of limits of vector functions and the fact that the component functions , , and are each differentiable we have,  



  

Example (Derivative of a Vector Function) Find the derivative of the vector function



    Solution. The derivative is the vector function  

vector functions
operations with vector functions
limits and continuity of vector functions
graphs of vector functions
vector differentiation
tangent vectors
smooth curves
derivative rules
vector integration
motion of an object
unit tangent and unit normal vectors
arc length function
curvature

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Vector Differentiation
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/vector-differentiation.html